Memories of Goro Shimura Don Blasius, Toni Bluher, Haruzo Hida, Kamal Khuri-Makdisi, Kenneth Ribet, Alice Silverberg, and Hiroyuki Yoshida Goro Shimura, a mathematician who greatly influenced number theory in the second half of the twentieth cen- tury, was born in Japan on February 23, 1930. Over a career spanning six decades, he repeatedly made transfor- mational discoveries that stimulated new lines of inves- tigation and played a central role in the development of the field. Shimura earned his degrees at the University of Tokyo and held appointments at the University of Tokyo and Osaka University. He was a professor at Princeton Uni- versity from 1964 until he retired in 1999. He authored numerous influential books and papers and was awarded a Guggenheim Fellowship in 1970, the Cole Prize in Num- ber Theory in 1977, the Asahi Prize in 1991, and the Steele Prize for Lifetime Achievement in 1996. Goro Shimura passed away in Princeton on May 3, 2019, at the age of eighty-nine. Don Blasius Goro Shimura advised my 1981 Princeton thesis and he was, through his research and guidance, the central figure of my intellectual life in graduate school and for many years afterwards. His ideas about how to do mathematics have influenced me throughout my career. Figure 1. Goro, Chikako, Haru, and Tomoko returning to the Arriving at Princeton in fall 1977, I had no plan to study US from Haneda Airport in 1971. number theory, had never read a book about it, and had never heard of Shimura, Iwasawa, Dwork, Langlands, or offering of an introductory course in algebraic number even Weil. Most graduate courses had incomprehensible theory stood out as a beacon of light. Algebra already course descriptions. In this context Professor Shimura’s had great appeal for me, and I was hooked right away by this perfect course. Later I took his courses on families of Don Blasius is Distinguished Professor of Mathematics at UCLA. His email abelian varieties, special values of 퐿-functions, period rela- address is [email protected]. tions, the arithmetic theory of automorphic forms, Eisen- Communicated by Notices Associate Editor William McCallum. stein series, theta functions, etc., all topics essential to his For permission to reprint this article, please contact: current research and, as it turned out, my initial research. [email protected]. He really taught for his students, not just to have a context DOI: https://doi.org/10.1090/noti2076 to explore a topic of interest to himself. Each course started MAY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 677 with minimal assumptions of background and went far end as his destination after the fact. In words, he told me into its subject with both clarity and an economy of means. he starts with a general idea but no fixed goal or conjecture. He always had his lecture fully written out in a notebook, Many people who did not know him have some imag- which would be on the table in front of him. He would lec- ination of Shimura as exclusively serious or severe. This ture without it except for complex formulas. For them, he is simply not true, and I mention two among many mo- would pick it up to refresh his memory, always explaining ments of wit that we both enjoyed. After I started a conver- clearly as he went. Shimura was always perfectly prepared sation with a remark about Hecke operators, he said, “The and engaging. first thing you say to me is always interesting.” And after By early in my second year, it was clear to me that I I complained to him that I found his paper on confluent wanted Shimura to supervise my dissertation. When I hypergeometric functions hard, he said, “You need to go queried him about it, he asked me to read his famous see an analyst!” 1971 text Introduction to the Arithmetic Theory of Automor- About other mathematicians, he freely acknowledged phic Functions. This book is a masterpiece of exposition, his debts to Eichler, Hecke, and Weil, who was a friend starting from scratch and giving, mostly with proofs, the for over four decades. He also had the highest respect for key themes of his research up to that point. For me it was Siegel. Once he told me, “His proofs are correct and you the perfect preparation for reading the articles on which can just use his theorems.” He did not think that about it was based. It was a transformative undertaking. As an many mathematicians. One day we discussed Weil’s 1967 anecdote, I mention that when I told him I had finished paper on the converse theorem. After a while he said, Chapter 3, the one on Hecke algebras and the relation with with intensity, “Weil is a genius.” I never heard him say 퐿-functions, he asked me how long I had spent on it. I told that about anyone else, even when discussing major works. him I had spent about a month, and he said that was “very About ideas, although he knew a great deal, he was some- fast” (meaning too fast). He was correct! Later that year, thing of a minimalist in his preferred way of writing. For when I asked to be his student formally, he agreed but im- example, when I referred to automorphic forms as sections posed the condition that I promise not to “fire” him. He of vector bundles, he queried me sharply as to whether explained that a student with whom he had been work- this language added anything. I had to admit that, in the ing for some time had just done that, and he was visibly context, except for curb appeal to some, it did not. As a hurt when he spoke about it. I told him that there was no consequence, I avoided such usage in my dissertation. chance of that. I was asked to make a brief summary concerning Shimura did not tend, at least with me, to engage in Shimura’s theory of canonical models and its antecedents. lengthy, detailed discussion of mathematics. Usually, af- In 1953, at the start of his career, Shimura created ([1]) ter a brief discussion of math, he would shift the con- the first theory of reduction mod 푝 of varieties in arbitrary versation to something else, ranging from departmental dimensions. In a December 1953 letter to Shimura, Weil gossip to Chinese stories. He did not give me a research called it “a very important step forward” and emphasized problem right away. Instead, he asked me to read articles. its promise for the further development of complex multi- The first was [3], mentioned below, and the second was plication. He also wrote that it was “just what is needed [to [17]. Sometime while reading the second paper, he gave study] modular functions of several variables.” The first me the simple-sounding research problem concerning its direction became the famous collaboration of Shimura main theme of relations between the periods of abelian and Yutaka Taniyama, which was well underway by 1955. varieties: “Find more precise results. Find natural fields of They defined and studied abelian (group) varieties ofCM definition.” This was a great problem because it connected type and proved the Shimura–Taniyama reciprocity law, with so many topics, including values of 퐿-functions and which describes explicitly the action of a Galois group on the extension of the canonical models formalism from au- the points of finite order of the variety. The main un- tomorphic functions to automorphic forms, both new ar- derlying result here is a celebrated formula for the prime eas. ideal decomposition of the endomorphism that reduces to Before giving me this problem, Shimura had supported the Frobenius morphism attached to a given prime of the my desire to come up with my own. Several times I made field of definition. As a key application of this formula, suggestions to him, but he found compelling issues with they computed the Hasse–Weil zeta function at almost all them, such as being too hard or already done. When I places, thereby verifying Hasse’s conjecture for such func- finally found some results on his problem, he washappy tions. Their research was summarized in the well-known and exclaimed “Isn’t it nice to have some success?!” Later, 1961 monograph Complex Multiplication of Abelian Vari- I asked him about how he worked. He drew a really messy eties and Its Applications to Number Theory, which Shimura self-intersecting path on the blackboard and declared the wrote after Taniyama’s death. In fact, Shimura wrote 678 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 5 research articles about complex multiplication and abelian varieties over his career, even publishing in 1998 an expos- itory monograph Abelian Varieties with Complex Multiplica- tion and Modular Functions. This text includes more recent fundamental topics of his research, such as reciprocity laws for values of modular functions at CM points and the the- ory of period relations for abelian varieties of CM type. From the late 1950s until the late 1960s Shimura made a continuing study, mostly published in the Annals of Math- ematics, of the fields of definition for certain varieties de- fined by arithmetic quotients of bounded symmetric do- mains. For me, the start was the 1963 article “On ana- Figure 2. Chikako and Goro in 1993 in Nagano prefecture, lytic families of polarized abelian varieties and automor- where they had a summer house. phic functions.” In its first part, this highly readable paper showed that arithmetic quotients of three of the four clas- symmetric domains I, II,” both published in the Annals.